How does an unsupervised learning model learn?
Supervised learning is typically an attempt to learn a mathematical function, $f(\bf X)=\bf y$. For this, you need both the input vector $\bf X$ and the output vector $\bf y$. The model outputs have whatever dimensionality that the target values have.

Unsupervised learning models instead learn a structure from the data. A clustering model, for example, is learning both how many clusters exist in the data (a number that's not the same type as the inputs) and where those clusters are located (which is also a different type from the inputs). The output of running this model on a new datapoint $x$ is not the same type as $x$, but instead a classification label.

Similarly, time series models learn parameters that symbolize how vectors in the input relate to each other, rather than raw inputs themselves.

As for how they learn, the structures are mathematical objects whose fitness is determined by the input data. The simplest possible unstructured unsupervised learning problem is probably "what's the mean of the data?", and it should be clear how that's 'learned' through processing the input. More sophisticated models are just adding more pieces to that calculation.